List of Group Constructions

Introduction

This article lists the constructions by which the corpus builds a group from smaller groups: the direct products and direct sums, the semidirect and wreath products, the extensions, the free products, the free products with amalgamation, the HNN extensions and the quotients by normal subgroups. For each construction the list records what it preserves — whether the factors remain as subgroups, whether the result is finite when the factors are, whether the factors commute — and the article that introduces it.

Every entry points to the article that introduces the construction. This article is a list: it introduces no definition, states no theorem, gives no proof, and carries no display mathematics. It records examples and non-examples side by side, a non-example being a construction that fails to have a property its neighbour has, with the failure named and the article that records it.

Direct Products and Direct Sums

The direct product of a family of groups is the group of functions into the family with pointwise multiplication; the direct sum is the subgroup of finitely supported functions, and for finitely many factors the two agree. The product is the categorical product with its projections, the sum the coproduct with its injections, and a group is the internal direct product of two subgroups when they commute and generate it.

Construction	What it preserves	Introduced in
Direct product $\prod_i G_i$	the projections; the factors embed and commute elementwise	Infinite Abelian Groups
Direct sum $\bigoplus_i G_i$	the injections; the finitely supported subgroup of the product	Infinite Abelian Groups
Direct power $G^I$	the product of $\|I\|$ copies of one group	Generators, Presentations and Free Products
Internal direct product	two commuting subgroups generating the group	Groups
Direct product of finitely many factors	finite when the factors are finite, of order the product of the orders	Groups
Non-example: $S_3 \times \mathbb{Z}/2\mathbb{Z}$	the direct product is not the semidirect product: the factors commute, and $S_3 \times \mathbb{Z}/2\mathbb{Z}$ has no normal $S_3$ complement	Groups

Semidirect and Wreath Products

The semidirect product $N \rtimes_\varphi H$ is built from a homomorphism $\varphi : H \to \operatorname{Aut}(N)$, so that $H$ acts on $N$ and the product law is twisted by $\varphi$; it is the split extension of $H$ by $N$. The wreath product $A \wr H$ is the semidirect product of a direct sum of copies of $A$ indexed by $H$ with $H$ acting by permutation of the summands.

Construction	What it preserves	Introduced in
Semidirect product $N \rtimes_\varphi H$	both factors as subgroups; $N$ normal, $H$ a complement	Generators, Presentations and Free Products
Split extension $1 \to N \to G \to H \to 1$ with section $s$	the short exact sequence splits; $G$ is a semidirect product	Generators, Presentations and Free Products
Dihedral group as $C_n \rtimes C_2$	the reflection action on the rotation group	Finite Groups and Symmetry
Affine group $\operatorname{Aff}(V) = V \rtimes GL(V)$	the translations normal, the linear group a complement	Affine Spaces and Translations
Wreath product $A \wr H = A^{(H)} \rtimes H$	the base group $A^{(H)}$ normal, $H$ permuting coordinates	Infinite Groups
Lamplighter group $(\mathbb{Z}/2\mathbb{Z}) \wr \mathbb{Z}$	a finitely generated metabelian group that is not Noetherian	Infinite Groups
Hyperoctahedral group $W(B_n) = S_n \ltimes (\mathbb{Z}/2\mathbb{Z})^n$	a wreath product realised as a reflection group	Symplectic Reflection Algebras (Part II)
Non-example: $N \rtimes_\varphi H$ with $\varphi$ nontrivial	fails to be the direct product: the factors do not commute	Generators, Presentations and Free Products
Non-example: $A \wr H$ for infinite $H$	fails to be finitely generated in general: the base group is an infinite direct sum	Infinite Groups

Extensions and Cohomological Constructions

An extension of $H$ by $N$ is a short exact sequence $1 \to N \to E \to H \to 1$; it is split when a section exists, so that $E$ is a semidirect product, and central when $N$ lies in the centre. The extensions inducing a fixed action of $H$ on $N$ are classified by the second cohomology $H^2(H,N)$, with the split extension as the zero class and the Baer sum as the addition; the third cohomology classifies the crossed modules and $2$-fold extensions.

Construction	What it preserves	Introduced in
Extension $1 \to N \to E \to H \to 1$	$N$ normal in $E$; the quotient isomorphic to $H$	Generators, Presentations and Free Products
Classification by $H^2(H,N)$	the action of $H$ on $N$ and the equivalence classes of extensions	Group Cohomology
Split extension	the extension has a section; equivalently the class is zero	Group Cohomology
Central extension	$N$ central in $E$; classified by $H^2(H,N)$ with trivial action	Group Cohomology
Universal central extension	through which every central extension of a perfect group factors uniquely	Group Cohomology
Schur multiplier $M(G) = H_2(G,\mathbb{Z})$	the kernel of the universal central extension	Group Cohomology
Schur–Zassenhaus splitting	the extension splits when $\gcd(\|M\|,\|G\|)=1$	Group Cohomology
Non-example: a nonsplit extension of coprime order	does not exist: Schur–Zassenhaus forces a splitting	Group Cohomology
Non-example: an extension of $H$ by $N$ with a nonabelian kernel	fails the $H^2$ classification: the kernel must be abelian for the cohomological classification	Group Cohomology

Free, Amalgamated and HNN Constructions

The free product $G_1 * G_2$ is the coproduct of groups, the universal group receiving both factors with no relation between them; it is infinite whenever one factor is nontrivial and has more than one element. The free product with amalgamation $G_1 *_H G_2$ identifies a common subgroup, and the HNN extension adjoins a stable letter conjugating one subgroup onto another; both are the group-theoretic pushout and its relative.

Construction	What it preserves	Introduced in
Free product $G_1 * G_2$	both factors as subgroups; no relation between them	Generators, Presentations and Free Products
Infinite dihedral group $D_\infty = C_2 * C_2$	a free product that is virtually cyclic	Generators, Presentations and Free Products
Free product with amalgamation $G_1 *_H G_2$	the common subgroup $H$; the pushout over it	Combinatorial Group Theory
HNN extension	a subgroup identified with its image under an isomorphism via a stable letter	Combinatorial Group Theory
Kurosh subgroup theorem	the subgroups of a free product are free products of copies of subgroups of the factors and a free group	Combinatorial Group Theory
Nielsen–Schreier theorem	every subgroup of a free group is free	Combinatorial Group Theory
Presentation $\langle X \mid R\rangle$	the quotient of the free group by the normal closure of $R$	Generators, Presentations and Free Products
Non-example: $G_1 * G_2$ for nontrivial $G_1, G_2$	fails to be finite unless both factors have order $2$: the free product is infinite	Generators, Presentations and Free Products
Non-example: $G_1 *_H G_2$ with $H$ not normal in either factor	fails the pushout description when the maps do not identify $H$	Combinatorial Group Theory

Quotients, Series and Presentations

A quotient $G/N$ is formed by a normal subgroup, and the quotient of a group by the normal closure of a set of relations is a presentation. The derived and lower central series are iterated quotients, and their termination defines the solvable and the nilpotent groups; the abelianisation is the universal abelian quotient.

Construction	What it preserves	Introduced in
Quotient $G/N$	the group law on the cosets; exists exactly for normal $N$	Groups
Normal closure $\langle\langle R\rangle\rangle$	the smallest normal subgroup containing a set of relations	Generators, Presentations and Free Products
Abelianisation $G^{\mathrm{ab}} = G/[G,G]$	the largest abelian quotient of $G$	Groups
Derived series	the iterated commutator subgroups; terminates for solvable groups	Solvable and Nilpotent Groups
Lower central series	the iterated commutators with $G$; terminates for nilpotent groups	Solvable and Nilpotent Groups
Subgroup generated by $S$	the smallest subgroup containing $S$	Groups
Non-example: $G/N$ for a non-normal $N$	fails to be a group: the coset multiplication is not well defined	Groups

What Each Construction Preserves

The constructions differ in the properties they preserve; the table collects the comparison that the scope line asks for.

Construction	Preserves finiteness	Preserves freeness	Keeps factors as subgroups
Direct product	yes, for finitely many factors	yes	yes
Direct sum	yes, for finitely many factors	yes	yes
Semidirect product	yes, for finite factors	only when the action is trivial	yes
Wreath product	yes, for finite factors	no	yes
Extension	yes, for finite kernel and quotient	no	the kernel only
Free product	no, unless both factors have order $2$	yes	yes
Free product with amalgamation	no, in general	no	the factors, modulo $H$
HNN extension	no, in general	no	the base group
Quotient	yes	no	no

Summary

The list gathers the group constructions of the corpus. The products are the direct product, the direct sum, the direct power and the internal direct product; the twisted products are the semidirect product, the wreath product, the lamplighter group, the dihedral group and the affine group; the extensions are the general extension, its classification by the second cohomology, the split and central extensions, the universal central extension with the Schur multiplier as kernel, and the Schur–Zassenhaus splitting; the free constructions are the free product, the free product with amalgamation, the HNN extension and the presentations, with the Kurosh and Nielsen–Schreier theorems; the quotients are the quotient by a normal subgroup, the abelianisation, the normal closure and the derived and lower central series. The final table compares what each construction preserves, and the non-examples — the direct product that is not a semidirect product, the semidirect product with a nontrivial action, the infinite wreath product, the coprimal-order nonsplit extension, the extension with a nonabelian kernel, the infinite free product, the free product with a non-identified subgroup and the quotient by a non-normal subgroup — name the property that fails.

Summary of Notation

The article denotes its constructions by name; the symbols appearing in the tables are those of the introducing articles.

Symbol	Meaning
$\prod_i G_i$, $\bigoplus_i G_i$, $G^I$	direct product, direct sum, direct power
$N \rtimes_\varphi H$	semidirect product with action $\varphi$
$A \wr H$	wreath product
$1 \to N \to E \to H \to 1$	short exact sequence, an extension
$H^n(G,M)$, $H_2(G,\mathbb{Z})$	group cohomology; Schur multiplier
$G_1 * G_2$	free product
$G_1 *_H G_2$	free product with amalgamation
$\langle X \mid R\rangle$, $\langle\langle R\rangle\rangle$	presentation and normal closure
$G^{\mathrm{ab}} = G/[G,G]$	abelianisation
$[G,G]$	commutator subgroup